Central Limit Theorem Calculator

The Central Limit Theorem states that when you draw sufficiently large random samples from any population, the distribution of the sample means will approximate a normal distribution, regardless of the population's underlying distribution shape. The core mathematical relationships governing this calculator are straightforward. First, the expected value of the sample mean equals the population mean (μ_x̄ = μ). Second, the variability of these sample means—known as the standard error (σ_x̄)—is calculated by taking the population standard deviation (σ) and dividing it by the square root of the sample size (n). The formula is expressed as σ_x̄ = σ / √n.

Imagine you are studying a massive university entrance exam where the population mean score (μ) is 100, and the population standard deviation (σ) is 15. You decide to draw random samples of n = 25 students. First, the expected sample mean remains identical to the population mean, meaning your expected sample mean is 100. Next, calculate the standard error by dividing the population standard deviation by the square root of the sample size: 15 / √25. Since the square root of 25 is 5, the standard error is 15 / 5 = 3. This means that while individual test scores vary widely, groups of 25 students will have average scores that cluster around 100 with a standard error of 3.

Always verify your sample size before applying CLT assumptions; while a sample size of 30 is widely accepted as the threshold for normality, heavily skewed populations may require larger samples. Ensure that your samples are selected independently and randomly to avoid selection bias, which can distort the true standard error. Finally, remember that the standard error decreases only as the square root of the sample size increases, meaning quadrupling your sample size is necessary to cut the standard error in half.